A binary set is a set with (exactly) two distinct elements, or, equivalently, a set whose cardinality is two. Examples:
- The set {a,b} is binary.
- The set {a,a} is not binary, since it is the same set as {a}, and is thus a singleton.
In axiomatic set theory, the existence of binary sets is a consequence of the axiom of empty set and the axiom of pairing. From the axiom of empty set it is known that the set <math>\emptyset = \{\}</math> exists. From the axiom of pairing it is then known that the set <math>\{\emptyset,\emptyset\} = \{\emptyset\}</math> exists, and thus the set <math>\{\{\emptyset\},\emptyset\}</math> exists. This latter set has two elements.
